Reynolds Number
Laminar vs Turbulent Flow — Formula, Critical Values, Velocity Profiles & Solved Problems
Last Updated: March 2026
📌 Key Takeaways
- Reynolds number: Re = ρVD/μ = VD/ν — ratio of inertial forces to viscous forces.
- Dimensionless — no units. Determines whether flow is laminar or turbulent.
- Pipe flow: Re < 2,300 → Laminar. Re > 4,000 → Turbulent. 2,300–4,000 → Transition.
- Laminar flow: Smooth, orderly, parabolic velocity profile, Vmax = 2Vavg.
- Turbulent flow: Chaotic, mixing, flatter velocity profile, much higher friction losses.
- Named after Osborne Reynolds, who demonstrated the transition experimentally in 1883.
1. The Formula — What It Means
Reynolds Number
Re = ρVD/μ = VD/ν
Where:
- ρ = fluid density (kg/m³)
- V = average flow velocity (m/s)
- D = characteristic length — pipe diameter for internal flow (m)
- μ = dynamic viscosity (Pa·s)
- ν = kinematic viscosity = μ/ρ (m²/s)
Re is dimensionless — it has no units.
2. Physical Interpretation
The Reynolds number represents the ratio of inertial forces to viscous forces in a fluid:
Re = Inertial forces / Viscous forces = ρV²/L / (μV/L²) = ρVL/μ
| Re Value | Dominant Force | Flow Character |
|---|---|---|
| Low Re (< 2,300) | Viscous forces dominate | Laminar — viscosity keeps flow orderly, damps disturbances |
| High Re (> 4,000) | Inertial forces dominate | Turbulent — inertia overwhelms viscosity, flow becomes chaotic |
Think of it this way: viscosity acts like glue holding fluid layers together. At low velocities (low Re), this glue is strong enough to maintain orderly layers. At high velocities (high Re), the fluid’s momentum (inertia) overpowers the glue, and layers break into chaotic eddies and swirls.
3. Laminar vs Turbulent Flow
| Feature | Laminar Flow | Turbulent Flow |
|---|---|---|
| Flow pattern | Smooth, parallel layers — no mixing | Chaotic eddies, swirls — intense mixing |
| Velocity profile | Parabolic (Vmax = 2Vavg) | Flatter (Vmax ≈ 1.2Vavg) |
| Reynolds number | Re < 2,300 (pipe flow) | Re > 4,000 (pipe flow) |
| Friction losses | Low — proportional to V | High — proportional to V1.75 to V² |
| Friction factor dependence | f = 64/Re (Hagen-Poiseuille) | f depends on Re AND pipe roughness (Moody chart) |
| Heat/mass transfer | Poor — transfer only by molecular diffusion | Excellent — turbulent mixing enhances transfer |
| Predictability | Highly predictable, exact solutions exist | Statistical/empirical — exact prediction impossible |
| Real-world examples | Blood flow in capillaries, slow oil flow, groundwater | Water in pipes, river flow, atmospheric winds, most engineering flows |
4. Critical Reynolds Number
| Geometry | Characteristic Length | Recritical |
|---|---|---|
| Circular pipe (internal flow) | Pipe diameter D | ~2,300 |
| Flat plate (external flow) | Distance from leading edge x | ~5 × 10⁵ |
| Flow over sphere/cylinder | Diameter D | ~2 × 10⁵ (sphere) |
| Open channel | Hydraulic radius Rh | ~500 |
The transition zone (2,300 < Re < 4,000 for pipes) is unstable. Small disturbances can trigger turbulence at Re = 2,300, but in very smooth pipes with minimal disturbances, laminar flow has been maintained up to Re ≈ 40,000 in laboratory conditions. For engineering design, assume transition at Re = 2,300 and fully turbulent above 4,000.
5. Velocity Profiles
Laminar Flow — Parabolic Profile
Hagen-Poiseuille Velocity Profile
u(r) = (ΔP / 4μL) × (R² − r²)
Where: u(r) = velocity at radial distance r from centre, R = pipe radius, L = pipe length, ΔP = pressure drop
Vmax = (ΔP × R²) / (4μL) — at the pipe centre (r = 0)
Vavg = Vmax / 2 — the average velocity is exactly half the maximum
Turbulent Flow — Flatter Profile
Turbulent mixing causes the velocity profile to be much flatter than parabolic. The power-law approximation is commonly used:
u/Vmax = (1 − r/R)1/n
Where n ≈ 7 for typical turbulent flow (the “1/7th power law”)
Vavg / Vmax ≈ 0.82 (for n = 7)
6. Laminar Flow in Pipes — Key Results
Hagen-Poiseuille Equation — Flow Rate
Q = πΔPD⁴ / (128μL)
Flow rate is proportional to D⁴ — doubling the diameter increases flow by 16 times (at same ΔP).
Pressure Drop — Laminar Pipe Flow
ΔP = 128μLQ / (πD⁴) = 32μLV / D²
Friction Factor — Laminar Flow
f = 64 / Re (Darcy friction factor)
This is an exact result, valid only for laminar flow (Re < 2,300).
Wall Shear Stress
τw = 8μV/D = ΔPD/(4L)
7. Reynolds Number for Other Geometries
For non-circular cross-sections, use the hydraulic diameter:
Hydraulic Diameter
Dh = 4A / Pwetted
Where A = cross-sectional area, Pwetted = wetted perimeter
| Cross-Section | Dh |
|---|---|
| Circular pipe (diameter D) | Dh = D |
| Square duct (side a) | Dh = a |
| Rectangular duct (a × b) | Dh = 2ab/(a+b) |
| Annulus (outer D₁, inner D₂) | Dh = D₁ − D₂ |
8. Worked Numerical Examples
Example 1: Determine Flow Type
Problem: Water (ν = 1 × 10⁻⁶ m²/s) flows through a 50 mm diameter pipe at 0.5 m/s. Is the flow laminar or turbulent?
Solution
Re = VD/ν = 0.5 × 0.05 / (1 × 10⁻⁶) = 0.025 / 10⁻⁶ = 25,000
Re = 25,000 > 4,000 → Turbulent flow
Example 2: Maximum Velocity for Laminar Flow
Problem: Oil with kinematic viscosity 4 × 10⁻⁴ m²/s flows through a 100 mm pipe. What is the maximum velocity for the flow to remain laminar?
Solution
For laminar flow: Re ≤ 2,300
VD/ν ≤ 2300 → V ≤ 2300ν/D = 2300 × 4 × 10⁻⁴ / 0.1
Vmax = 9.2 m/s
Oil’s high viscosity allows laminar flow at much higher velocities than water.
Example 3: Laminar Pipe Flow — Pressure Drop
Problem: Oil (μ = 0.1 Pa·s, ρ = 900 kg/m³) flows through a 25 mm diameter, 50 m long pipe at Q = 0.5 L/s. Find the pressure drop and verify the flow is laminar.
Solution
Q = 0.5 × 10⁻³ m³/s, D = 0.025 m, A = π(0.025)²/4 = 4.909 × 10⁻⁴ m²
V = Q/A = 0.5 × 10⁻³ / 4.909 × 10⁻⁴ = 1.018 m/s
Re = ρVD/μ = 900 × 1.018 × 0.025 / 0.1 = 229 → Laminar ✓
ΔP = 128μLQ/(πD⁴) = 128 × 0.1 × 50 × 0.5 × 10⁻³ / (π × 0.025⁴)
= 3.2 / (π × 3.906 × 10⁻⁷) = 3.2 / (1.227 × 10⁻⁶)
ΔP = 2,607,987 Pa ≈ 2,608 kPa ≈ 26.1 bar
High pressure drop despite low Re — viscous oil in a small pipe requires significant pumping pressure.
9. Common Mistakes Students Make
- Using pipe radius instead of diameter: Re = VD/ν uses pipe DIAMETER, not radius. Using radius gives Re that is half the correct value, potentially misidentifying the flow regime.
- Confusing dynamic and kinematic viscosity: Re = ρVD/μ (with dynamic viscosity) or Re = VD/ν (with kinematic viscosity). Mixing them up gives dimensionally wrong results.
- Applying f = 64/Re to turbulent flow: This formula is valid ONLY for laminar flow (Re < 2,300). For turbulent flow, use the Moody chart or Colebrook equation.
- Forgetting Vavg = Vmax/2 for laminar flow: In laminar pipe flow, the maximum centreline velocity is exactly twice the average velocity. Exam problems often give one and ask for the other.
- Not using hydraulic diameter for non-circular ducts: For rectangular, annular, or other cross-sections, replace D with Dh = 4A/P in the Reynolds number formula.
10. Frequently Asked Questions
What is the Reynolds number?
The Reynolds number is a dimensionless quantity (Re = ρVD/μ) that predicts the flow regime — laminar or turbulent. It represents the ratio of inertial forces (which promote chaos) to viscous forces (which promote order). Low Re means viscous forces win (smooth laminar flow); high Re means inertia wins (chaotic turbulent flow).
What is the critical Reynolds number for pipe flow?
For internal flow in circular pipes, Recritical ≈ 2,300. Below this, flow is laminar; above ~4,000, it is turbulent. The range 2,300–4,000 is the transition zone. For external flow over a flat plate, the critical value is much higher: Recritical ≈ 5 × 10⁵.
What is the difference between laminar and turbulent flow?
Laminar flow is smooth and predictable — fluid moves in parallel layers with a parabolic velocity profile. Turbulent flow is chaotic — fluid particles mix randomly with a flatter velocity profile. Turbulent flow has higher friction losses (more energy wasted) but also much better heat and mass transfer (more mixing). Most engineering flows at practical velocities are turbulent.